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Reducing the complexity of frequency response masking filters using half-band filters
PROCESSING ELSEVIER
Signal Processing 42 (1995) 227-230
Reducing the complexity of frequency response masking filters using half-band filters Yong Lian, Yong Ching Lim* Department
of Electrical Engineen’ng, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511. Singapore
Received 19 November 1993; revised 5 May 1994
Abstract
The frequency-response masking approach provides an efficient realization for very sharp FIR filters with arbitrary bandwidth. In this paper, we introduce a method which uses a half-band filter to serve as one of the masking filters to adhieve further saving in the number of multipliers. Zusammenfawmg Das Frequenzgang-Maskierungsverfahren liefert eine effiziente Realisierung von sehr steilflankigen FIR-Filtern mit beliebiger Bandbreite. In dieser Arbeit fiihren wir eine Methode ein, welche ein Halbbandfilter als Maskierungsfilter benutzt und so eine weitere Reduktion der Multiplikationsanzahl herbeifiihrt.
L’approche par masquage de la rkponse en frbquence permet d’obtenir une rtalisation efficace de filtres FIR g flancs t&s raides de largeur de bande arbitraire. Nous introduisons dans cet article une mCthode qui utilise un filtre demi-bande comme l’un des filtres de masquage afin d’obtenir une rkduction additionnelle du nombre de multiplications. Keywords: Frequency-response masking; Half-band filter; Sparse coefficient filter; Sharp filter; Narrow transition width filter; Low complexity FIR digital filter
1. Introduction
Many methods have been proposed to reduce the number of multipliers required for implementing sharp FIR filters in the past few years. One of the most attractive methods for the synthesis of arbitrary bandwidth sharp FIR filters is the
*Corresponding author. 0165-1684/95/S9.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0165-1684(94)00131-6
frequency-response masking approach introduced in [cl] and subsequently developed further in I2 4, 51. Given a prototype symmetrical impulse response linear phase low-pass filter H,(z) of odd length N,, its complementary filter H,(z) can be expressed as H,(z) = z-(Na- 1)/Z- H,(z).
(1)
Replacing each delay elements of both H,(z) and H,(z) by A4delays, two filters with transfer functions
228
Y. Lian. Y.C. Lim 1 Signal Processing 42 (1995) 227-230
passband and stopband edges, respectively [3]. In the frequency-response masking approach, the band-edge of H(z) is determined jointly by Hya(z) and H,,(z) as shown in Figs. l(b) and l(c). The case corresponding to Fig. l(b) is referred to as CASE A and that corresponding to Fig. l(c) is referred to as CASE B. Let F, and F, be the normalized passband and stopband edges of H(z). It can be shown that for CASE A, F p~a+Fs~a+P=--,
2m + 1 M
F p~r+F~~r-8=~r
(34
(W
and for CASE B, F p~a+Fs~a-P=-, 0
K
Fig. 1. The frequency responses of the various subfilters used in frequency-response masking approach.
and H&Z?) are formed. The transition widths of H,(z) and H,(z) are a factor of M narrower than that of H,(z). In the frequency-response masking technique, two filters, Hya(z) and H,,(z), are cascaded to H,(z”) and H,(z”), respectively. The outputs of H,(z”)Hicr,(z) and H,(z”)HMJz) are then summed to form H(z): H,(z”)
H(z) = K(z”)&i(4 + (Z-M=
1)/Z- H,(zM))&(z).
(2)
Note that the group delay of H,,(z) and Hyc(z) must be equal, and M(N, - 1) must be even [l]. The frequency responses of the above approach are shown in Fig. 1. In this paper, we introduce a method which makes use of a half-band filter to serve as one of the masking filters. This leads to additional saving in the number of multipliers.
2. The masking filter It is a well-known fact that for a half-band filter f, -tfs = 0.5, where fP and fs are the normalized
F p~r+F~~.+~=~~
2n - 1 M
(3c) (34
In (3) Fp~a and Fs~a are the passband and stopband edges of H,,(z), respectively; FpMc and FsMc are the passband and stopband edges of H&z), respectively; /l = F, - F,; m = LF,M 1, where LXJ denotes the largest integer smaller than or equal to X; n = r F,M 1, where [xl denotes the smallest integer larger than or equal to x. In order to ensure that one of the masking filters will satisfy the half-band filter requirement, a suitable value of M should be selected in such a way that the right-hand side of one of (3a)-(3d) is equal to 0.5. However, such a value of M may not be the optimum M derived in [2]. Nevertheless, in many cases, the total number of multipliers required for implementing H(z) is decreased because about half of the number of multipliers of a half-band filter has zero value. It is possible to achieve extra saving in the number of multipliers by relaxing the stopband edge of the masking filters very slightly. From Fig. l(b), it can be seen that if the stopband edge of H,,(z) is relaxed slightly into the transition band of H,(z”), the attenuation at the stopband of H(z) corresponding to the overlap between the transition bands of H~Jz) and H,(z”) will be degraded. This degradation will manifest itself as a ripple in the stopband of H(z). The magnitude of this ripple can be
Y. Lian, Y.C. Lim / Signal Processing
constrained to any acceptable value as follows. If 6(w) is the frequency response deviation of H(ejW), then ignoring the second order term,
O.OT-----l -20.0
d(m) = (6~M,(w)-s,,(w))H,(ej”M) + (HMM,(ejW) - H~,(e’“)P,(w) + &,(c&
(4)
where 6,(w), 6,,(o) and 6,,(o) are the frequency response deviations of Ha(ejwM), H,&e@) and HMMc(ejW), respectively. Thus, as can be seen from (4) the magnitude of 6(o) can be controlled by controlling the value of Ha(ejWM),HMa(ejU)and HMMc(ejO). If two of the three functions, H,(ejaM), HMMa(e@) and HMM,(ejU), are fixed and the third one is optimized to minimize for the maximum value of lS(o)l, such an optimization problem is a linear optimization problem and can be performed using standard algorithm such as the linear programming algorithm.
229
42 (1995) 227-230
k?
\
I
-40.0 -
j
-00.0 -
Normalized Frequency
Fig. 2. The frequency response of the overall system.
3. The design procedure A procedure for the synthesis of our new structure is presented below. This procedure is a modified version of that reported in [2]. Step 1. Evaluate Mopt = (4fi)-“2. Choose a value of M which is approximately equal to M,,, and at the same time will make the right-hand side of one of (3a)-(3d) equal to 0.5 Step 2. Design two masking filters H,,(z) and H,,(z). One of the masking filters should be a halfband filter. Step 3. Use a standard FIR filter design program to design H,(z) taking the above two masking filters as prefilters. Detailed design formulas can be found in [ 11. If the stopband edge of the half-band filter is relaxed into the transition band of H,(P), go to Step 4; otherwise, stop. Step 4. Taking H,(z) and the half-band masking filter as prefilters, redesign the other masking filter. Step 5. Redesign H,(z) using the two masking filters as prefilters.
4. An example We shall illustrate our method by using an example. Consider the design of a low-pass FIR
filter with band-edges at 0.23 and 0.235 sampling frequencies. The passband and stopband ripples are the same and are equal to 0.01. With M = 5, the total number of multipliers and group delay required by the filter designed using the method reported in [l] is 62 and 208, respectively. Using this new method with M = 10, the filter length of H,(z), H,,(z) and H,,(z) are 39, 55 and 29, respectively. The masking filter H&z) is a half-band filter. The total number of multipliers required for this design is 49, and the group delay is 217. Comparing to the design obtained by using the method reported in Cl], our new technique produces an extra saving of 13 multipliers at the expense of an increase in the group delay by 9 samples. The frequency response of the filter is shown in Fig. 2.
5. Conclusion In this paper, a modified frequency-response masking approach is presented. The success of our approach is due to the use of a half-band filter to serve as one of the masking filters. This leads to a further saving in the number of multipliers. A modified design procedure for the frequency response masking approach is also derived.
230
Y. Lian, Y.C. Lim / Signal Processing 42 (1995) 227-230
References [l] Y.C. Lim, “Frequency-response masking approach for the synthesis of sharp linear phase digital filter”, IEEE Trans. CAS, Vol. 33, No. 4, April 1986, pp. 357-364. [2] Y.C. Lim and Y. Lian, “The optimum design of one- and two-dimensional FIR filters using the frequency-response masking technique”, IEEE Trans. CAS Part II, Vol. 40, No. 2, February 1993, pp. 88-95.
[3] F. Mintzer, “On half-band, third-band, and Nth-band FIR filters and their design”, IEEE Trans. Acoust. Speech Signal Process., Vol. 30, No. 5, October 1982, pp. 734-738. [4] G. Rajan, Y. Neuvo and S.K. Mitra, “On the design of sharp cutoff wide-band FIR filters with reduced arithmetic complexity”, IEEE Trans. CAS, Vol. 3.5, No. 11, November 1988, pp. 1447-1454. [S] R. Yang, B. Liu and Y.C. Lim, “A new structure of sharp transition FIR filters using frequency-response masking”, IEEE Trans. CAS, Vol. 35, No. 8, August 1988,pp. 955-966.
Signal Processing 42 (1995) 227-230
Reducing the complexity of frequency response masking filters using half-band filters Yong Lian, Yong Ching Lim* Department
of Electrical Engineen’ng, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511. Singapore
Received 19 November 1993; revised 5 May 1994
Abstract
The frequency-response masking approach provides an efficient realization for very sharp FIR filters with arbitrary bandwidth. In this paper, we introduce a method which uses a half-band filter to serve as one of the masking filters to adhieve further saving in the number of multipliers. Zusammenfawmg Das Frequenzgang-Maskierungsverfahren liefert eine effiziente Realisierung von sehr steilflankigen FIR-Filtern mit beliebiger Bandbreite. In dieser Arbeit fiihren wir eine Methode ein, welche ein Halbbandfilter als Maskierungsfilter benutzt und so eine weitere Reduktion der Multiplikationsanzahl herbeifiihrt.
L’approche par masquage de la rkponse en frbquence permet d’obtenir une rtalisation efficace de filtres FIR g flancs t&s raides de largeur de bande arbitraire. Nous introduisons dans cet article une mCthode qui utilise un filtre demi-bande comme l’un des filtres de masquage afin d’obtenir une rkduction additionnelle du nombre de multiplications. Keywords: Frequency-response masking; Half-band filter; Sparse coefficient filter; Sharp filter; Narrow transition width filter; Low complexity FIR digital filter
1. Introduction
Many methods have been proposed to reduce the number of multipliers required for implementing sharp FIR filters in the past few years. One of the most attractive methods for the synthesis of arbitrary bandwidth sharp FIR filters is the
*Corresponding author. 0165-1684/95/S9.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0165-1684(94)00131-6
frequency-response masking approach introduced in [cl] and subsequently developed further in I2 4, 51. Given a prototype symmetrical impulse response linear phase low-pass filter H,(z) of odd length N,, its complementary filter H,(z) can be expressed as H,(z) = z-(Na- 1)/Z- H,(z).
(1)
Replacing each delay elements of both H,(z) and H,(z) by A4delays, two filters with transfer functions
228
Y. Lian. Y.C. Lim 1 Signal Processing 42 (1995) 227-230
passband and stopband edges, respectively [3]. In the frequency-response masking approach, the band-edge of H(z) is determined jointly by Hya(z) and H,,(z) as shown in Figs. l(b) and l(c). The case corresponding to Fig. l(b) is referred to as CASE A and that corresponding to Fig. l(c) is referred to as CASE B. Let F, and F, be the normalized passband and stopband edges of H(z). It can be shown that for CASE A, F p~a+Fs~a+P=--,
2m + 1 M
F p~r+F~~r-8=~r
(34
(W
and for CASE B, F p~a+Fs~a-P=-, 0
K
Fig. 1. The frequency responses of the various subfilters used in frequency-response masking approach.
and H&Z?) are formed. The transition widths of H,(z) and H,(z) are a factor of M narrower than that of H,(z). In the frequency-response masking technique, two filters, Hya(z) and H,,(z), are cascaded to H,(z”) and H,(z”), respectively. The outputs of H,(z”)Hicr,(z) and H,(z”)HMJz) are then summed to form H(z): H,(z”)
H(z) = K(z”)&i(4 + (Z-M=
1)/Z- H,(zM))&(z).
(2)
Note that the group delay of H,,(z) and Hyc(z) must be equal, and M(N, - 1) must be even [l]. The frequency responses of the above approach are shown in Fig. 1. In this paper, we introduce a method which makes use of a half-band filter to serve as one of the masking filters. This leads to additional saving in the number of multipliers.
2. The masking filter It is a well-known fact that for a half-band filter f, -tfs = 0.5, where fP and fs are the normalized
F p~r+F~~.+~=~~
2n - 1 M
(3c) (34
In (3) Fp~a and Fs~a are the passband and stopband edges of H,,(z), respectively; FpMc and FsMc are the passband and stopband edges of H&z), respectively; /l = F, - F,; m = LF,M 1, where LXJ denotes the largest integer smaller than or equal to X; n = r F,M 1, where [xl denotes the smallest integer larger than or equal to x. In order to ensure that one of the masking filters will satisfy the half-band filter requirement, a suitable value of M should be selected in such a way that the right-hand side of one of (3a)-(3d) is equal to 0.5. However, such a value of M may not be the optimum M derived in [2]. Nevertheless, in many cases, the total number of multipliers required for implementing H(z) is decreased because about half of the number of multipliers of a half-band filter has zero value. It is possible to achieve extra saving in the number of multipliers by relaxing the stopband edge of the masking filters very slightly. From Fig. l(b), it can be seen that if the stopband edge of H,,(z) is relaxed slightly into the transition band of H,(z”), the attenuation at the stopband of H(z) corresponding to the overlap between the transition bands of H~Jz) and H,(z”) will be degraded. This degradation will manifest itself as a ripple in the stopband of H(z). The magnitude of this ripple can be
Y. Lian, Y.C. Lim / Signal Processing
constrained to any acceptable value as follows. If 6(w) is the frequency response deviation of H(ejW), then ignoring the second order term,
O.OT-----l -20.0
d(m) = (6~M,(w)-s,,(w))H,(ej”M) + (HMM,(ejW) - H~,(e’“)P,(w) + &,(c&
(4)
where 6,(w), 6,,(o) and 6,,(o) are the frequency response deviations of Ha(ejwM), H,&e@) and HMMc(ejW), respectively. Thus, as can be seen from (4) the magnitude of 6(o) can be controlled by controlling the value of Ha(ejWM),HMa(ejU)and HMMc(ejO). If two of the three functions, H,(ejaM), HMMa(e@) and HMM,(ejU), are fixed and the third one is optimized to minimize for the maximum value of lS(o)l, such an optimization problem is a linear optimization problem and can be performed using standard algorithm such as the linear programming algorithm.
229
42 (1995) 227-230
k?
\
I
-40.0 -
j
-00.0 -
Normalized Frequency
Fig. 2. The frequency response of the overall system.
3. The design procedure A procedure for the synthesis of our new structure is presented below. This procedure is a modified version of that reported in [2]. Step 1. Evaluate Mopt = (4fi)-“2. Choose a value of M which is approximately equal to M,,, and at the same time will make the right-hand side of one of (3a)-(3d) equal to 0.5 Step 2. Design two masking filters H,,(z) and H,,(z). One of the masking filters should be a halfband filter. Step 3. Use a standard FIR filter design program to design H,(z) taking the above two masking filters as prefilters. Detailed design formulas can be found in [ 11. If the stopband edge of the half-band filter is relaxed into the transition band of H,(P), go to Step 4; otherwise, stop. Step 4. Taking H,(z) and the half-band masking filter as prefilters, redesign the other masking filter. Step 5. Redesign H,(z) using the two masking filters as prefilters.
4. An example We shall illustrate our method by using an example. Consider the design of a low-pass FIR
filter with band-edges at 0.23 and 0.235 sampling frequencies. The passband and stopband ripples are the same and are equal to 0.01. With M = 5, the total number of multipliers and group delay required by the filter designed using the method reported in [l] is 62 and 208, respectively. Using this new method with M = 10, the filter length of H,(z), H,,(z) and H,,(z) are 39, 55 and 29, respectively. The masking filter H&z) is a half-band filter. The total number of multipliers required for this design is 49, and the group delay is 217. Comparing to the design obtained by using the method reported in Cl], our new technique produces an extra saving of 13 multipliers at the expense of an increase in the group delay by 9 samples. The frequency response of the filter is shown in Fig. 2.
5. Conclusion In this paper, a modified frequency-response masking approach is presented. The success of our approach is due to the use of a half-band filter to serve as one of the masking filters. This leads to a further saving in the number of multipliers. A modified design procedure for the frequency response masking approach is also derived.
230
Y. Lian, Y.C. Lim / Signal Processing 42 (1995) 227-230
References [l] Y.C. Lim, “Frequency-response masking approach for the synthesis of sharp linear phase digital filter”, IEEE Trans. CAS, Vol. 33, No. 4, April 1986, pp. 357-364. [2] Y.C. Lim and Y. Lian, “The optimum design of one- and two-dimensional FIR filters using the frequency-response masking technique”, IEEE Trans. CAS Part II, Vol. 40, No. 2, February 1993, pp. 88-95.
[3] F. Mintzer, “On half-band, third-band, and Nth-band FIR filters and their design”, IEEE Trans. Acoust. Speech Signal Process., Vol. 30, No. 5, October 1982, pp. 734-738. [4] G. Rajan, Y. Neuvo and S.K. Mitra, “On the design of sharp cutoff wide-band FIR filters with reduced arithmetic complexity”, IEEE Trans. CAS, Vol. 3.5, No. 11, November 1988, pp. 1447-1454. [S] R. Yang, B. Liu and Y.C. Lim, “A new structure of sharp transition FIR filters using frequency-response masking”, IEEE Trans. CAS, Vol. 35, No. 8, August 1988,pp. 955-966.